2017/11/05 by Ivan Dungan, Dungan, Ivan
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1711.01638
openalex publication_date 2017/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Crossed modules are known to be a model of pointed connected homotopy 2-types; formally, the homotopy category of crossed modules is equivalent to the category of pointed connected homotopy 2-types. In forming the homotopy category of crossed modules, one must resort to computing derived morphisms using non-constructive topological methods, but Behrang Noohi was able to find an algebraic model of these derived morphisms called butterflies. The result is a completely algebraic model of pointed connected homotopy 2-types. Reduced crossed complexes are a generalization of crossed modules and model a subclass of pointed connected homotopy types. We will present algebraic objects called n-butterflies which satisfy similar properties to butterflies and begin to generalize the theory of butterflies to model morphisms of pointed connected homotopy n-types.